Finite Group Databasebeta

← C5×#63925 #13087197

Subgroups — C5×#63925 #13087197

Order 1440.

294 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#237C211yesmaximal, characteristic, complemented
2C5×D3×D12C211yesmaximal, characteristic, complemented
3C5×#3753C211yesmaximal, characteristic, complemented
4C3×C5×#213C211yesmaximal, characteristic, complemented
5C5×#3717C211yesmaximal, characteristic, complemented
6C5×#3715C211yesmaximal, characteristic, complemented
7C5×#3711C211yesmaximal, characteristic, complemented
8C5×#949—31nomaximal
9C5×#911—31nomaximal
10C2×C5×D32—21no—
11C32×C5×D4C2211yescharacteristic, complemented
12C5×C4⋊(C3⋊D3)C2211yescharacteristic
13C3×C5×#67—21no—
14C2×C3×C5×Dic3—21no—
15C3×C5×D12C2211yescharacteristic
16C3×C4×C5×D3C2211yescharacteristic
17C3×C5×Dic6C2211yescharacteristic, complemented
18C5×#626—21no—
19C5×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
20C3×C5×C3⋊C8C2211yescharacteristic, complemented
21#63925C511yesmaximal, characteristic, direct product factor, Hall
22C3×C5×#41—31no—
23C5×#237D311yescharacteristic, complemented
24C5×D3×D4—31no—
25C2×C5×D12—31no—
26C3×C5×D8—31no—
27C5×#215—31no—
28C5×#215—31no—
29C5×#214—31no—
30C5×#213—31no—
31C5×#213—31no—
32C5×#213—61no—
33C5×#213D311yescharacteristic, complemented
34C5×#208—31no—
35C5×#204—31no—
36C5×ΓL(1,𝔽25)—31no—
37C22×C32×C5—21no—
38C2×C5×C3⋊D3—21no—
39C2×C3×C5×D3—21no—
40C2×C3×C5×D3D411yescharacteristic
41C5×D32—42no—
42C32×C4×C5C2311yescharacteristic
43C3×C5×Dic3—21no—
44C3×C5×Dic3D411yescharacteristic
45C5×AGL(1,ℤ/8)—91noHall
46C3×#237C2×C511yescharacteristic, complemented
47D3×D12C2×C511yescharacteristic, complemented
48#3753C2×C511yescharacteristic, complemented
49C3×#213C2×C511yescharacteristic, complemented
50#3717C2×C511yescharacteristic, complemented
51#3715C2×C511yescharacteristic, complemented
52#3711C2×C511yescharacteristic, complemented
53C22×C5×D3—61no—
54C22×C5×D3—61no—
55C3×C5×Q8—31no—
56C3×C5×D4—21no—
57C3×C5×D4—31no—
58C3×C5×D4—61no—
59C3×C5×D4C2×D311yescharacteristic, complemented
60C3×C5×D4C2×D311yescharacteristic, complemented
61C2×C3×C4×C5—31no—
62C2×C3×C4×C5—61no—
63C5×#67—21no—
64C5×#67—61no—
65C2×C5×Dic3—21no—
66C5×D12—31no—
67C5×D12—31no—
68C5×D12—61no—
69C5×D12—61no—
70C5×D12C2×D311yescharacteristic
71C4×C5×D3C2×D311yescharacteristic
72C5×Dic6C2×D311yescharacteristic, complemented
73C3×C5×C8—31no—
74C5×C3⋊C8—31no—
75C5×C3⋊C8—31no—
76C5×C3⋊C8—61no—
77C5×C3⋊C8C2×D311yescharacteristic, complemented
78#949—31no—
79#911—31no—
80C2×C32×C5—41no—
81C2×C32×C5C2×D411yescharacteristic
82C5×C3⋊D3—41no—
83C3×C5×D3—21no—
84C3×C5×D3—41no—
85C5×#41—31no—
86C2×C5×D4—91no—
87C5×SD16—91no—
88C5×SD16—91no—
89C5×D8—31no—
90C5×D8—91no—
91C5×M16—91no—
92C2×D32—21no—
93C32×D4C22×C511yescharacteristic, complemented
94C4⋊(C3⋊D3)C22×C511yescharacteristic
95C3×#67—21no—
96C2×C3×Dic3—21no—
97C3×D12C22×C511yescharacteristic
98C3×C4×D3C22×C511yescharacteristic
99C3×Dic6C22×C511yescharacteristic, complemented
100#626—21no—
101C3⋊(C3⋊C8)C22×C511yescharacteristic, complemented
102C3×C3⋊C8C22×C511yescharacteristic, complemented
103C22×C3×C5—21no—
104C22×C3×C5—21no—
105C22×C3×C5—31no—
106C22×C3×C5—41no—
107C22×C3×C5—61no—
108C2×C5×D3—21no—
109C2×C5×D3—61no—
110C2×C5×D3—61no—
111C2×C5×D3—121no—
112C2×C5×D3—122no—
113C2×C5×D3—122no—
114C2×C5×D3#6711yescharacteristic
115C3×C4×C5—21no—
116C3×C4×C5—31no—
117C3×C4×C5—61no—
118C3×C4×C5C22×D311yescharacteristic
119C3×C4×C5C22×D311yescharacteristic
120C5×Dic3—21no—
121C5×Dic3#6711yescharacteristic
122C3×#41—31no—
123#237C5×D311yescharacteristic, complemented
124D3×D4—31no—
125C2×D12—31no—
126C3×D8—31no—
127#215—31no—
128#215—31no—
129#214—31no—
130#213—31no—
131#213—31no—
132#213—61no—
133#213C5×D311yescharacteristic, complemented
134#208—31no—
135#204—31no—
136ΓL(1,𝔽25)—31no—
137C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
138C23×C5—181no—
139C5×Q8—31no—
140C5×D4—31no—
141C5×D4—61no—
142C5×D4—91no—
143C5×D4—181no—
144C5×D4D3211yescharacteristic, complemented
145C2×C4×C5—31no—
146C2×C4×C5—61no—
147C5×C8—31no—
148C5×C8—91no—
149C22×C32—21no—
150C2×C3⋊D3—21no—
151C2×C3×D3—21no—
152C2×C3×D3C5×D411yescharacteristic
153D32—42no—
154C32×C4C23×C511yescharacteristic
155C3×Dic3—21no—
156C3×Dic3C5×D411yescharacteristic
157AGL(1,ℤ/8)—91noSylow
158C2×C3×C5—21no—
159C2×C3×C5—41no—
160C2×C3×C5—41no—
161C2×C3×C5—61no—
162C2×C3×C5—81no—
163C2×C3×C5—121no—
164C2×C3×C5D3×D411yescharacteristic
165C2×C3×C5C2×#6711yescharacteristic
166C5×D3—21no—
167C5×D3—41no—
168C5×D3—121no—
169C5×D3—121no—
170C5×D3—241no—
171C22×D3—61no—
172C22×D3—61no—
173C3×Q8—31no—
174C3×D4—21no—
175C3×D4—31no—
176C3×D4—61no—
177C3×D4C2×C5×D311yescharacteristic, complemented
178C3×D4C2×C5×D311yescharacteristic, complemented
179C2×C3×C4—31no—
180C2×C3×C4—61no—
181#67—21no—
182#67—61no—
183C2×Dic3—21no—
184D12—31no—
185D12—31no—
186D12—61no—
187D12—61no—
188D12C2×C5×D311yescharacteristic
189C4×D3C2×C5×D311yescharacteristic
190Dic6C2×C5×D311yescharacteristic, complemented
191C3×C8—31no—
192C3⋊C8—31no—
193C3⋊C8—31no—
194C3⋊C8—61no—
195C3⋊C8C2×C5×D311yescharacteristic, complemented
196C22×C5—21no—
197C22×C5—31no—
198C22×C5—61no—
199C22×C5—181no—
200C22×C5—362no—
201C4×C5—31no—
202C4×C5—61no—
203C4×C5C2×D3211yescharacteristic
204C2×C32—41no—
205C2×C32C2×C5×D411yescharacteristic
206C3⋊D3—41no—
207C3×D3—21no—
208C3×D3—41no—
209#41—31no—
210C2×D4—91no—
211SD16—91no—
212SD16—91no—
213D8—31no—
214D8—91no—
215M16—91no—
216C3×C5—21no—
217C3×C5#91111yescharacteristic, complemented
218C3×C5#94911yescharacteristic, complemented
219C22×C3—21no—
220C22×C3—21no—
221C22×C3—31no—
222C22×C3—41no—
223C22×C3—61no—
224C2×D3—21no—
225C2×D3—61no—
226C2×D3—61no—
227C2×D3—121no—
228C2×D3—122no—
229C2×D3—122no—
230C2×D3C5×#6711yescharacteristic
231C3×C4—21no—
232C3×C4—31no—
233C3×C4—61no—
234C3×C4C22×C5×D311yescharacteristic
235C3×C4C22×C5×D311yescharacteristic
236Dic3—21no—
237Dic3C5×#6711yescharacteristic
238C2×C5—41no—
239C2×C5—61no—
240C2×C5—121no—
241C2×C5—361no—
242C2×C5D3×#6711yescharacteristic
243C32C5×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
244C23—181no—
245Q8—31no—
246D4—31no—
247D4—61no—
248D4—91no—
249D4—181no—
250D4C5×D3211yescharacteristic, complemented
251C2×C4—31no—
252C2×C4—61no—
253C8—31no—
254C8—91no—
255C2×C3—21no—
256C2×C3—41no—
257C2×C3—41no—
258C2×C3—61no—
259C2×C3—81no—
260C2×C3—121no—
261C2×C3C5×D3×D411yescharacteristic
262C2×C3C2×C5×#6711yescharacteristic
263D3—21no—
264D3—41no—
265D3—121no—
266D3—121no—
267D3—241no—
268C5#6392511yescharacteristic, direct product factor, Sylow
269C22—21no—
270C22—31no—
271C22—61no—
272C22—181no—
273C22—362no—
274C4—31no—
275C4—61no—
276C4C2×C5×D3211yescharacteristic
277C3—21no—
278C3C5×#91111yescharacteristic, complemented
279C3C5×#94911yescharacteristic, complemented
280C2—41no—
281C2—61no—
282C2—121no—
283C2—361no—
284C2C5×D3×#6711yescharacteristic
subgroupG/H
centerC2×C5D3×#67
commutator, cosocleC32×C4C23×C5
FrattiniC4C2×C5×D32
FittingC32×C5×D4C22
socleC2×C32×C5C2×D4
hypercenterC5×D4D32
hypocenterC32C5×AGL(1,ℤ/8)
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C5 ◁ C4×C5 ◁ C5×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×C5×#237C2semidirect
2. C5×D3×D12C2semidirect
3. C5×#3753C2semidirect
4. C3×C5×#213C2semidirect
5. C5×#3717C2semidirect
6. C5×#3715C2semidirect
7. C5×#3711C2semidirect
11. C32×C5×D4C22semidirect
17. C3×C5×Dic6C22semidirect
19. C5×C3⋊(C3⋊C8)C22semidirect
20. C3×C5×C3⋊C8C22semidirect
21. #63925C5direct (N × H)
23. C5×#237D3semidirect
33. C5×#213D3semidirect
46. C3×#237C2×C5semidirect
47. D3×D12C2×C5semidirect
48. #3753C2×C5semidirect
49. C3×#213C2×C5semidirect
50. #3717C2×C5semidirect
51. #3715C2×C5semidirect
52. #3711C2×C5semidirect
59. C3×C5×D4C2×D3semidirect
60. C3×C5×D4C2×D3semidirect
72. C5×Dic6C2×D3semidirect
77. C5×C3⋊C8C2×D3semidirect
93. C32×D4C22×C5semidirect
99. C3×Dic6C22×C5semidirect
101. C3⋊(C3⋊C8)C22×C5semidirect
102. C3×C3⋊C8C22×C5semidirect
123. #237C5×D3semidirect
133. #213C5×D3semidirect
137. C32×C5AGL(1,ℤ/8)semidirect
144. C5×D4D32semidirect
177. C3×D4C2×C5×D3semidirect
178. C3×D4C2×C5×D3semidirect
190. Dic6C2×C5×D3semidirect
195. C3⋊C8C2×C5×D3semidirect
217. C3×C5#911semidirect
218. C3×C5#949semidirect
243. C32C5×AGL(1,ℤ/8)semidirect
250. D4C5×D32semidirect
268. C5#63925direct (N × H)
278. C3C5×#911semidirect
279. C3C5×#949semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×C5×#2372
C5×D3×D122
C5×#37532
C3×C5×#2132
C5×#37172
C5×#37152
C5×#37112
C5×#9493
C5×#9113
#639255